The Homogenized Energy Model (HEM) for Characterizing Polarization and Strains in Hysteretic Ferroelectric Materials: Material Properties and Uniaxial Model Development

Ferroelectric materials, such as PZT, PLZT and BaTiO3, are being considered, or are already being employed, for a large number of applications including nanopositioning, high speed valves for fuel injectors, ultrasonic transducers, high speed camera shutters and auto focusing mechanisms, energy harvesting, and pico air vehicle design. Their advantages include nanometer positioning resolution, broadband frequency responses, moderate power requirements, the capability for miniaturization, and complementary actuator and sensor capabilities. However, they also exhibit creep, rate-dependent hysteresis, and constitutive nonlinearities at essentially all drive levels due to their noncentrosymmetric nature. In this paper, we model the hysteretic dependence of strains and polarization on input fields and stresses using the homogenized energy model (HEM) framework. At the domain level, the minimization of Gibbs energy densities yields linear constitutive relations. Nonlinearities and hysteresis due to dipole switching is modeled at the grain level by using Boltzmann theory to specify the evolution of dipole fractions which serve as internal variables. In the final step of the development, stochastic homogenization, based on the assumption that interaction fields and driving forces are manifestations of underlying densities, is used to construct nonlinear constitutive relations for the bulk material. It is demonstrated that these relations are amenable to subsequent development of distributed system models. The paper includes significant discussion regarding the mechanisms that produce hysteresis in ferroelectric materials. The capability of the framework for characterizing various hysteretic phenomena, including creep and various rate-dependencies, is illustrated by validation with PZT and PLZT data.